3.1.51 \(\int (e+f x)^m (a+b \tanh ^{-1}(c+d x))^2 \, dx\) [51]

Optimal. Leaf size=23 \[ \text {Int}\left ((e+f x)^m \left (a+b \tanh ^{-1}(c+d x)\right )^2,x\right ) \]

[Out]

Unintegrable((f*x+e)^m*(a+b*arctanh(d*x+c))^2,x)

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Rubi [A]
time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int (e+f x)^m \left (a+b \tanh ^{-1}(c+d x)\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(e + f*x)^m*(a + b*ArcTanh[c + d*x])^2,x]

[Out]

Defer[Subst][Defer[Int][((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcTanh[x])^2, x], x, c + d*x]/d

Rubi steps

\begin {align*} \int (e+f x)^m \left (a+b \tanh ^{-1}(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int \left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^m \left (a+b \tanh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ \end {align*}

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Mathematica [A]
time = 0.23, size = 0, normalized size = 0.00 \begin {gather*} \int (e+f x)^m \left (a+b \tanh ^{-1}(c+d x)\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(e + f*x)^m*(a + b*ArcTanh[c + d*x])^2,x]

[Out]

Integrate[(e + f*x)^m*(a + b*ArcTanh[c + d*x])^2, x]

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Maple [A]
time = 2.37, size = 0, normalized size = 0.00 \[\int \left (f x +e \right )^{m} \left (a +b \arctanh \left (d x +c \right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^m*(a+b*arctanh(d*x+c))^2,x)

[Out]

int((f*x+e)^m*(a+b*arctanh(d*x+c))^2,x)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^m*(a+b*arctanh(d*x+c))^2,x, algorithm="maxima")

[Out]

1/4*(b^2*f*x + b^2*e)*(f*x + e)^m*log(-d*x - c + 1)^2/(f*(m + 1)) + (f*x + e)^(m + 1)*a^2/(f*(m + 1)) - integr
ate(-1/4*((b^2*d*f*(m + 1)*x + (c*f*(m + 1) - f*(m + 1))*b^2)*log(d*x + c + 1)^2 + 4*(a*b*d*f*(m + 1)*x + (c*f
*(m + 1) - f*(m + 1))*a*b)*log(d*x + c + 1) - 2*(b^2*d*e + 2*(c*f*(m + 1) - f*(m + 1))*a*b + (2*a*b*d*f*(m + 1
) + b^2*d*f)*x + (b^2*d*f*(m + 1)*x + (c*f*(m + 1) - f*(m + 1))*b^2)*log(d*x + c + 1))*log(-d*x - c + 1))*(f*x
+ e)^m/(d*f*(m + 1)*x + c*f*(m + 1) - f*(m + 1)), x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^m*(a+b*arctanh(d*x+c))^2,x, algorithm="fricas")

[Out]

integral((b^2*arctanh(d*x + c)^2 + 2*a*b*arctanh(d*x + c) + a^2)*(f*x + e)^m, x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**m*(a+b*atanh(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^m*(a+b*arctanh(d*x+c))^2,x, algorithm="giac")

[Out]

integrate((b*arctanh(d*x + c) + a)^2*(f*x + e)^m, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int {\left (e+f\,x\right )}^m\,{\left (a+b\,\mathrm {atanh}\left (c+d\,x\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e + f*x)^m*(a + b*atanh(c + d*x))^2,x)

[Out]

int((e + f*x)^m*(a + b*atanh(c + d*x))^2, x)

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